Properties

Degree 16
Conductor $ 3^{16} \cdot 5^{16} \cdot 7^{8} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·16-s − 8·31-s − 4·41-s + 8·61-s − 16·71-s + 81-s − 8·101-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 2·16-s − 8·31-s − 4·41-s + 8·61-s − 16·71-s + 81-s − 8·101-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1575} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.07132271713\)
\(L(\frac12)\)  \(\approx\)  \(0.07132271713\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
7 \( 1 - T^{4} + T^{8} \)
good2 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
17 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
19 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
31 \( ( 1 + T + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
43 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
59 \( ( 1 - T^{2} + T^{4} )^{4} \)
61 \( ( 1 - T + T^{2} )^{8} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T )^{16} \)
73 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
89 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
97 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.05167961364493317466960213993, −4.03797993435311022904855569012, −3.92881235075408855005909353873, −3.91792195142930063995557640940, −3.91134506199235148467388215251, −3.70719740993966932509469020224, −3.38515846241032667959857457831, −3.16003554654582166684090138636, −3.15398381557730017306994834060, −3.09969212426919849260878773271, −2.99856500217957436810707241741, −2.93760197554213190169260213367, −2.93678566040578532679537013896, −2.64313276421559857091952309918, −2.27961541601106450202869188209, −1.98491876706737872351358226189, −1.96924782360785571842396142702, −1.90488381638609017449048181607, −1.80503919206670076590381488672, −1.62785111165918555781214206912, −1.57445781972414155785619589187, −1.25884491901131651590652254621, −1.23447959103052203842511443275, −0.855545169668131933687633899687, −0.11204262707789757538186926695, 0.11204262707789757538186926695, 0.855545169668131933687633899687, 1.23447959103052203842511443275, 1.25884491901131651590652254621, 1.57445781972414155785619589187, 1.62785111165918555781214206912, 1.80503919206670076590381488672, 1.90488381638609017449048181607, 1.96924782360785571842396142702, 1.98491876706737872351358226189, 2.27961541601106450202869188209, 2.64313276421559857091952309918, 2.93678566040578532679537013896, 2.93760197554213190169260213367, 2.99856500217957436810707241741, 3.09969212426919849260878773271, 3.15398381557730017306994834060, 3.16003554654582166684090138636, 3.38515846241032667959857457831, 3.70719740993966932509469020224, 3.91134506199235148467388215251, 3.91792195142930063995557640940, 3.92881235075408855005909353873, 4.03797993435311022904855569012, 4.05167961364493317466960213993

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.